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0.20.1 0.3
a) 3-class plan: n=5, c=0
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
0.1 0.1
0.2 0.3
b) 3-class plan: n=5, cm=1, cM=0
0.1 0.3 0.5 0.7 0.9
c) 3-class plan: n=5, cm=2, cM=0
0.1 0.5 0.3 0.80.7 0.9
d) 3-class plan: n=5, cm=3,cM=0 pd:proportion defective in lot
0.0 0.2 0.4 0.6 0.8 1.0
pd:proportion defective in lot
0.0 0.2 0.4 0.6 0.8 1.0
pd:proportion defective in lot
pm
m
m m
: proportion marginally acceptable in lot
0.0 0.2 0.4 0.6 0.8 1.0
p : proportion marginally acceptable in lot
0.0 0.2 0.4 0.6 0.8 1.0
p : proportion marginally acceptable in lot
0.0 0.2 0.4 0.6 0.8 1.0
pd:proportion defective in lot
0.0 0.2 0.4 0.6 0.8 1.0
p : proportion marginally acceptable in lot Fig. 7.4 Contour maps of operating characteristic function of 3-class attributes plan for sample size n = 5 and different criteria of acceptance(c). Numbers within graphs (e.g. 0.1, 0.2, 0.3 in graph a) represent probablity of acceptance
function should be plotted as an OC surface, either in a three- dimensional graph, or as a contour map in the two-dimensional (pm, pd) area with contour lines for selected acceptance probabilities. Such OC function maps are shown in Fig. 7.4 for three-class sampling plans with n = 5 and different acceptance numbers c = 0, c = 1, c = 2, and c = 3.
All lots with combinations of pm and pd lying on the same contour line in such a graph have the same probability of acceptance that is indicated at the end of the line. If, for instance, the three-class plan n = 5, c = 1 is applied, all kinds of lots with (pm, pd) combinations on the outermost line are accepted with a probability of only 0.1 or 10% of the times such a lot will be examined. Thus, the three-class attributes scheme is affected to some extent by the frequency distribution of microorgan-isms within the batch, but the advantages of the scheme are its simplicity and general applicability, which make it appropriate to port-of-entry sampling.
However, there is a need to elaborate sound methods to set the values of m and M. These should be related to actual concentrations of microorganisms and the frequency distributions of analytical results. There are statistically-based techniques for achieving this, although assumptions must be made. An example, based upon assumptions that can readily be checked and found (historically) rea-sonable, is Dahms and Hildebrandt (1998), which is explained in more detail in Sect. 7.4.4.
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7.3.1 Identification
To define a variables plan, the underlying frequency distribution for the microorganisms of concern within the sample units in a lot must be known. There are many types of frequency distributions, and they differ in complexity. An overview of aspects of distributions for microorganisms in foods is described by ILSI (Basset et al. 2010). Some of the simpler distributions are symmetrical in shape and can be described by their mean and some measure of the distribution about that mean. The normal, or Gaussian, distribution is one such example. The normal distribution is defined by its mean value (which is also the median) for the range of concentrations found, and a measure (σ, the standard deviation), which defines the possibility of finding any other concentration in a unit. More complex distributions may not be symmetrical, so that the mean is unlikely to equal the median.
What is essential to apply a variables plan is knowledge of frequency distribution. To achieve this, data may be required to support the application of a particular distribution. The more complex the frequency distribution (i.e., more parameters are needed to define it), the more data are needed to gain confidence that it is an appropriate distribution. An example of a distribution often observed for microorganisms in foods, the log-normal distribution, is used below (Sect. 7.3.3). It is worth noting that the scale has been adjusted to obtain a normal distribution. This adjustment results in a symmetri-cal, two parameter (mean and standard deviation) distribution, achieved by using log (concentration) instead of concentration.
It is important to remember that any measurement of the parameters for a distribution is based upon a sample and is therefore an estimate of those parameters. These measured parameters must accommodate the uncertainty implicit in the measurement. The smaller the sample size n, the larger the likely error could be. The example given in Sect. 7.3.3 illustrates this principle by allowing for sample size in its decision matrix.
7.3.2 Prescribing Confidence in Decisions
Making critical microbiological decisions about the safety or quality of a lot of food involves three steps. The first is to define the acceptable limits for the lot, the second is to specify the confidence with which we wish to identify acceptable and unacceptable lots, and the third is to choose the appropriate sampling plan. The following is an example of the way in which a variables plan may be designed. In this case, the decision rule is based upon an assumption that the underlying distribution of contami-nants in the lot is log-normal (i.e., the log of the concentrations is normally distributed). While this assumption is often correct, in practice, its justification needs to be clear and recorded. Assuming a log-normal distribution, sampling plans based on the characteristics of this distribution can be used to develop acceptance sampling plans.
7.3.3 Operation
It is necessary to obtain and handle samples and sample units in the same way as for attributes plans.
The log-transformation of the concentration measurements is used to compute the sample mean (x) and standard deviation (s). These two values are then used to make the decision whether to accept or reject the lot. The lot is rejected if x + k1s > V, where V is a log-concentration related to safety/quality limits.
The value k1 is obtained by reference to appropriate tables and is chosen to define the stringency of the plan for a given number of sample units, n.
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Selection of k1 Table 7.5 contains a range of k1 values for sample unit numbers between 3 and 10 (Malcolm 1984). To choose k1 it is necessary to decide on the maximum proportion pd of the units in the lot that can be accepted with a concentration above the limit value, V. Having selected pd, the desired probability P can be chosen, where P is the probability of rejecting a lot which contains at least a proportion pd above V.
For example, if five sample units are analysed per lot, then the k1 value can be chosen from Table 7.5. If a lot in which 10% of sample units exceeded V is to be rejected with a probability of 0.95, then the k1 value 3.4 would be used.
In practice, the two values pd and P will be selected along with the value V. The scheme then allows n to be selected over the range 3 to 10. The larger n becomes, the lower the chance of rejection of an acceptable lot.
Selection of the limit value V The limit value V is selected by the microbiologist as the safety or qual-ity limit, expressed as log-concentration. This value is likely to be numerically very similar to the value M used in the three-class attributes plans (Sect. 7.2.2).
Table 7.6 gives the results for the aerobic plate count (APC) analyses of five sample units obtained from a lot of poultry. An appropriate variables sampling plan might be P = 0.90, pd = 0.25, with a limit value of V = 7. The k1 value, obtained from Table 7.6, is 1.7. Applying the formula x + k1s, gives 5.039 + 1.7 x 0.378, which equals 5.682. This value is less than the limit value of 7, and the lot is therefore accepted.
The use of variables plan for good manufacturing practice Food producers often find it advantageous to specify a Good Manufacturing Practice (GMP) standard. It may be possible to apply the variables plan under these circumstances, applying the formula outlined previously. The criterion is to accept the lot if x + k2s < V. The k2 value for the GMP plan is obtained from Table 7.7. The values P and pd
are selected as before and the appropriate k2 value is obtained. The limit value, V, will be very similar numerically to the limit value, m, used in the three- class attribute plan.
For a more extensive treatment of the variables plan topic see Kilsby (1982), Kilsby et al. (1979), Malcolm (1984) and FAO/WHO (2016). The first three references describe the approach as described here without assuming a specific standard deviation, while in FAO/WHO (2016) the approach is explained for a given assumed standard deviation.
Table 7.5 k1 values calculated using the non-central t-distribution – safety/quality specification (reject if x + k1s > V) Number of sample units n
Probability (P) of rejection Proportion (pd) exceeding V 3 4 5 6 7 8 9 10
0.95 0.05 7.7 5.1 4.2 3.7 3.4 3.2 3.0 2.9
0.1 6.2 4.2 3.4 3.0 2.8 2.6 2.4 2.4
0.3 3.3 2.3 1.9 1.6 1.5 1.4 1.3 1.3
0.90 0.1 4.3 3.2 2.7 2.5 2.3 2.2 2.1 2.1
0.25 2.6 2.0 1.7 1.5 1.4 1.4 1.3 1.3
Values from Malcolm (1984)
x sample mean, V log-concentration related to safety/quality limits
Table 7.6 An example of aerobic plate counts for a sample of poultry (n = 5)
APC log10(APC) Mean log (x) Standard deviation (s)
40,000 4.602
69,000 4.839
81,000 4.909 5.039 0.378
200,000 5.301
350,000 5.544
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